How can we say time varying Electric field is inducing time varying magnetic field?

Maxwell's equations (Ampere's law and Faraday's Law) say that:

$$\nabla \times \vec{E} = - \frac{\partial B}{\partial t}$$

To me, the derivative on the LHS is a spatial one, the partial on the RHS is a time varying one. So all we can deduce from this is that a time varying magnetic field generates a curly E field (says nothing about time varying E field). So why do textbooks typically make the statement that time varying E field produces a time varying B field ?

1 Answer

There is

$$\nabla \times \vec{B} = \mu_0 \big( \vec{J} + \epsilon_0 \frac{\partial \vec{E}}{\partial t} \big)$$

as well.

• Yes, but this is saying a time varying E field generates a curly magnetic field. We don't know if the time varying magnetic field is necessary curly, it could just be time varying, does it not generate an E field then ? Feb 12, 2018 at 3:23
• Try differentiating this equation with $t$. See if/how to get a $\vec{B}$ that is curly but not time dependent. Feb 12, 2018 at 3:28